\(P, Q, R\) are any points on the sides \(BC, CA, AB\) respectively of the triangle \(ABC\). Prove that the circles \(AQR, BRP, CPQ\) meet in a point. \par Using a special case of this theorem, shew how to construct points \(L\) and \(M\) such that the angles \(LBC, LCA, LAB\) are equal, and the angles \(MCB, MAC, MBA\) are equal.
Find the angle between the lines given by the equation \[ ax^2+2hxy+by^2=0, \] and obtain the equation of the pair of lines bisecting the angles between the lines. Hence write down the equation of the most general pair of lines having the same bisectors.
Prove that the centre of the rectangular hyperbola which passes through four concyclic points \(A, B, C, D\) lies on the perpendicular drawn from the mid-point of the line joining any two of the four points to the line joining the remaining two points. \par Explain how the asymptotes can be constructed.
A line \(l\) is drawn through \(O\), the orthocentre of a triangle \(ABC\) and meets \(BC, CA, AB\) in \(D, E, F\) respectively. \(AO, BO, CO\) meet the circumcircle of \(ABC\) in \(P, Q, R\) respectively. Prove that \(DP, EQ, FR\) meet in a point \(S\) of the circumcircle, and that the parabola with \(S\) as focus and \(l\) as directrix touches \(BC, CA, AB\).
Prove that one conic confocal with \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) can be drawn to touch a general line. \par Tangents are drawn from a point \((h, k)\) to this system of confocal conics. Prove that the locus of points of contact is \[ \frac{x}{y-k} + \frac{y}{x-h} = \frac{a^2-b^2}{hy-kx}. \]
The polars of \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) with respect to the conic \(ax^2+by^2=1\) meet this conic in \(Q_1R_1\) and \(Q_2R_2\) respectively. Shew that the six points \(P_1Q_1R_1P_2Q_2R_2\) lie on a conic, say \(K\). \par If \(P_1\) is kept fixed, and \(P_2\) describes a line, prove that the conics \(K\) all pass through a fixed point on this line.
Prove that on a straight line there is in general one pair of points conjugate with regard to all the conics through four points \(A, B, C, D\). Hence, or otherwise, shew that, if \(D\) is the orthocentre of the triangle \(ABC\), then all the conics are rectangular hyperbolas.
Obtain the equation of the circumcircle of the triangle formed by the three lines \[ ax+by+c=0, \quad a'x+b'y+c'=0, \quad a''x+b''y+c''=0. \] Find also the equation of the circle which has this triangle as a self-conjugate triangle.
Prove that in polar coordinates \((r, \theta)\) the radius of curvature of a curve is given by \[ \frac{\left\{r^2 + \left(\frac{dr}{d\theta}\right)^2\right\}^{3/2}}{r^2+2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}. \] Prove that the radius of curvature of the curve \(r=a(1-\cos\theta)\) is \(\frac{4a}{3}\sin\frac{\theta}{2}\). Sketch the curve.
Prove that in general a system of coplanar forces can be reduced to a force acting at an assigned point in the plane together with a couple \(G\). \par \(ABCD\) is a square of side \(a\). Forces \(1,2,3,4,P,kP\) act in \(AB, BC, CD, DA, AC, DB\) respectively. Shew that the locus of a point which moves so that \(G\) is constant is a straight line which passes through the same point \(H\) in \(BD\) whatever the value of \(k\). Determine the ratio in which \(BD\) is divided by \(H\) when \(G=\frac{1}{2}aP\sqrt{2}\) in the sense \(ABCD\).